Newton Law in DGP Brane-World with Semi-Infinite Extra Dimension

Newton potential for DGP brane-world scenario is examined when the extra dimension is semi-infinite. The final form of the potential involves a self-adjoint extension parameter α, which plays a role of an additional mass (or distance) scale. The striking feature of Newton potential in this setup is that the potential behaves as seven-dimensional in long range when α is nonzero. For small α there is an intermediate range where the potential is five-dimensional. Five-dimensional Newton constant decreases with increase of α from zero. In the short range the four-dimensional behavior is recovered. The physical implication of this result is discussed in the context of the accelerating behavior of universe. ∗email: [email protected] †e-mail: [email protected] ‡e-mail: [email protected] 1 Although the extra dimensional theories have their own long history [1–3], the recent activities on this field seem to be motivated from string theories [4]. Much cosmological implications are investigated and established from the recent brane-world scenario. Especially Randall-Sundrum(RS) scenario [5,6], one of the recent brane-world scenario developed by making use of the warped extra dimensions, provides an clue for the nature of 1/r-type Newton potential in our universe. The original RS computation of Newton potential is extended and developed from various aspects [7–11]. Especially, Refs. [10,11] derived Newton potential arising due to the confined gravity on the brane when bulk space is a single copy of AdS5 from an aspect of the singular quantum mechanics(SQM). In this case the gravitational fluctuation equation is treated as an usual Schrödinger equation with a singular potential and it should be solved with incorporation of the self-adjoint extension technique [12]. The real parameter, say ξ, introduced in the course of the self-adjoint extension parametrizes the boundary condition(BC) the gravitational fluctuation obeys on the brane. The SQM approach is ,more recently, applied to the RS scenario when 4d induced gravity is involved [13,14]. The physical origin of the 4d induced gravity is one-loop quantum effect [15–17]. When ξ = 1/2 which makes a singular brane to be usual RS brane, the 4d induced term generates an intermediate range in which the 5d potential 1/r emerges. Furthermore, the other singular branes corresponding to different ξ may trap a massive graviton, leading to Yukawa-like gravitational behavior. Recently, the brane-world scenario with Minkowski bulk has attracted attention, which is often referred as DGP model [18,19]. The model also involves a 4d induced term and recently applied to the cosmological constant hierarchy and the accelerating universe [20–24]. In this letter we will examine Newton law assigning on the general 3-brane in DGP scenario when the extra dimension is infinite briefly and semi-infinite in detail. For the case of the infinite extra dimension it is well-known that Newton potential is fivedimensional at long range and four-dimensional at short range [18]. We will reproduce this result by applying SQM to the fluctuation equation. For the case of the semi-infinite extra 2 dimension the final form of Newton potential involves a self-adjoint extension parameter α, which makes an additional distance scale when α 6= 0. When α = 0, Newton potential is similar to that for the case of the infinite extra dimension. However, the 5d Newton constant G5 becomes G ′ 5 = 2G5 where G5 is 5d Newton constant derived when the extra dimension is infinite. The most striking result occurs in the long-range behavior of the potential when α > 0. In this range Newton potential becomes seven-dimensional. In the intermediate range the potential is five-dimensional with smaller Newton constant than 2G5. The four-dimensional potential is recovered at short range. Let us start with the Einstein-Hilbert action S = M ∫